Subgroups — #27193189

Order 13184.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20268168	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₀₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	#335×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₂₄	—	2	1	no	—
5	C₂×D₈₂₄	C₄	1	2	yes	—
6	#15693018	—	2	1	no	—
7	C₂×C₁₀₃⋊C₁₆	—	2	1	no	—
8	C₄⋊C₈×C₁₀₃	—	2	1	no	—
9	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
10	C₂×D₄₁₂	—	2	2	no	—
11	C₂×D₄₁₂	—	4	1	no	—
12	C₂×D₄₁₂	—	4	1	no	—
13	D₈₂₄	—	4	2	no	—
14	D₈₂₄	—	4	1	no	—
15	D₈₂₄	—	4	1	no	—
16	C₁₀₃⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₀₃	—	4	1	no	—
18	C₂×C₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₃	D₄	1	1	yes	characteristic
20	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
21	C₂²×D₁₀₃	—	4	2	no	—
22	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₀₃	SD₁₆	1	1	yes	characteristic
25	D₄₁₂	—	4	2	no	—
26	D₄₁₂	—	4	2	no	—
27	D₄₁₂	—	8	2	no	—
28	D₄₁₂	—	8	1	no	—
29	D₄₁₂	—	8	1	no	—
30	C₈×C₁₀₃	—	2	1	no	—
31	C₈×C₁₀₃	—	8	1	no	—
32	C₈×C₁₀₃	D₈	1	1	yes	characteristic
33	C₈×C₁₀₃	SD₁₆	1	1	yes	characteristic
34	C₂²×C₁₀₃	#102	1	1	yes	characteristic
35	C₂×D₁₀₃	—	8	2	no	—
36	C₂×D₁₀₃	—	8	2	no	—
37	C₂×D₁₀₃	—	8	2	no	—
38	C₄×C₁₀₃	—	2	1	no	—
39	C₄×C₁₀₃	#100	1	1	yes	characteristic
40	C₄×C₁₀₃	#102	1	1	yes	characteristic
41	C₄×C₁₀₃	D₁₆	1	1	yes	characteristic
42	C₄×C₁₀₃	SD₃₂	1	1	yes	characteristic
43	C₂×C₁₀₃	#331	1	1	yes	characteristic
44	C₂×C₁₀₃	#357	1	1	yes	characteristic
45	C₂×C₁₀₃	#361	1	1	yes	characteristic
46	D₁₀₃	—	16	2	no	—
47	#1361	—	103	1	no	maximal, Sylow
48	C₁₀₃	#1361	1	1	yes	characteristic, complemented, Sylow
49	#493	—	103	1	no	—
50	C₄⋊C₁₆	—	103	1	no	—
51	#335	D₁₀₃	1	1	yes	characteristic, complemented
52	C₂×D₈	—	103	2	no	—
53	C₂×D₈	—	206	1	no	—
54	#127	—	206	1	no	—
55	C₂×C₁₆	—	206	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
58	C₂×D₄	—	412	1	no	—
59	C₂×D₄	—	412	1	no	—
60	C₂×D₄	—	206	2	no	—
61	D₈	—	412	2	no	—
62	D₈	—	412	1	no	—
63	D₈	—	412	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	C₄×D₁₀₃	1	1	yes	characteristic
66	C₂×C₈	D₄₁₂	1	1	yes	characteristic
67	C₄²	C₂²⋊D₁₀₃	1	1	yes	characteristic
68	C₁₆	—	412	1	no	—
69	C₂³	—	412	2	no	—
70	D₄	—	824	2	no	—
71	D₄	—	824	1	no	—
72	D₄	—	824	1	no	—
73	D₄	—	412	2	no	—
74	D₄	—	412	2	no	—
75	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	#13110090	1	1	yes	characteristic
77	C₂×C₄	#13110092	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	8	1	no	—
80	C₈	D₈₂₄	1	1	yes	characteristic
81	C₈	#13110094	1	1	yes	characteristic
82	C₂²	—	824	2	no	—
83	C₂²	—	824	2	no	—
84	C₂²	—	824	2	no	—
85	C₂²	#15692941	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#15692937	1	1	yes	characteristic
88	C₄	#15692943	1	1	yes	characteristic
89	C₄	#15692962	1	1	yes	characteristic
90	C₄	#15692964	1	1	yes	characteristic
91	C₂	—	1648	2	no	—
92	C₂	#20267534	1	1	yes	characteristic
93	C₂	#20267591	1	1	yes	characteristic
94	C₂	#20267603	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692941
commutator	C₂×C₈×C₁₀₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₃
Fitting	#335×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#102
prime radical	#20268168	C₂
cosocle	C₄×C₈×C₁₀₃	C₂²
hypercenter	#335	D₁₀₃
hypocenter	C₁₀₃	#1361
LCS	G ▷ C₂×C₈×C₁₀₃ ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄⋊(C₁₀₃⋊C₁₆)	C₂	semidirect
3. #335×C₁₀₃	C₂	semidirect
48. C₁₀₃	#1361	semidirect
51. #335	D₁₀₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20268168	2
C₄⋊(C₁₀₃⋊C₁₆)	2
#335×C₁₀₃	2
#1361	103